SOLUTION: Are there any *paid* calculus tutors online? I need to hire one for a few questions. The derivative of a function at a point a is given by lim (f(a+h) � f(a)) f� (a) =h-

SOLUTION: Are there any paid calculus tutors online? I need to hire one for a few questions. The derivative of a function at a point a is given by lim (f(a+h) � f(a)) f� (a) =h-

Question 870605: Are there any *paid* calculus tutors online? I need to hire one for a few questions.
The derivative of a function at a point a is given by

lim (f(a+h) � f(a))
f� (a) =h->0
h
(that's not showing up right. It is f'(a) = lim/h is approaching 0; then parentheses with f, all over h)
Explain where this limit comes from and provide meaning to what �f�(a)� represents. It is encouraged to interpret this limit geometrically/graphically.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
This is from the definition of the derivative

$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$

where x = a. So instead of calculating f'(x), which is a general algebraic expression, you are calculating f'(a) which is a single number.

Geometrically, or graphically, f'(a) is the slope of the tangent line to f(x) at the point x = a on the function f(x). Put another way, we have some tangent line y = mx+b where m = f'(a) and the tangent line goes through the point (a, f(a)).

Here is a good animation that shows what's going on.

http://math.bu.edu/people/tkohl/teaching/spring2013/secant.html

Visually, h is the horizontal distance from point P to Q. The secant line is going through points P,Q.
As Q gets closer to P, that secant line is slowly becoming a tangent line.
It is only a tangent line when P = Q since tangent lines only cross the function (locally) at one point.
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